1
1.0

Sep 18, 2013
09/13

by
Tom Sanders

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Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times log n as n tends to infinity.

Source: http://arxiv.org/abs/math/0605519v2

3
3.0

Sep 21, 2013
09/13

by
Tom Sanders

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3

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Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.

Source: http://arxiv.org/abs/1011.0107v2

4
4.0

Sep 23, 2013
09/13

by
Tom Sanders

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We prove the following result due to Hamidoune using an analytic approach. Suppose that A is a subset of a finite group G with |AA^{-1}| \leq (2-\varepsilon)|A|. Then there is a subgroup H of G and a set X of size O_\varepsilon(1) such that A \subset XH.

Source: http://arxiv.org/abs/1212.0457v1

3
3.0

Sep 23, 2013
09/13

by
Tom Sanders

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We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-\epsilon} for all \epsilon>0.

Source: http://arxiv.org/abs/0710.0642v4

12
12

Sep 23, 2013
09/13

by
Tom Sanders

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In this article we survey some of the recent developments in the structure theory of set addition.

Source: http://arxiv.org/abs/1212.0458v1

7
7.0

Jul 24, 2013
07/13

by
Tom Sanders

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We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).

Source: http://arxiv.org/abs/0807.5101v2

2
2.0

Sep 18, 2013
09/13

by
Tom Sanders

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Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+\epsilon}}.

Source: http://arxiv.org/abs/math/0605520v2

4
4.0

Jul 20, 2013
07/13

by
Tom Sanders

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We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).

Source: http://arxiv.org/abs/1007.5444v2

3
3.0

Sep 18, 2013
09/13

by
Tom Sanders

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We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - \epsilon} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.

Source: http://arxiv.org/abs/math/0605522v2

3
3.0

Sep 23, 2013
09/13

by
Tom Sanders

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We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d.

Source: http://arxiv.org/abs/0710.2545v2

36
36

Jul 24, 2013
07/13

by
Tom Sanders

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36

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We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|.

Source: http://arxiv.org/abs/0807.5100v2

6
6.0

Sep 20, 2013
09/13

by
Tom Sanders

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We prove a theorem claimed in math.CA/0605519 which asserts that if A is a subset of a compact abelian group G with density of a particular (natural, although technical) form then the A(G)-norm (that is the sum of the absolute values of the Fourier transform) of the characteristic function of A cannot be too small.

Source: http://arxiv.org/abs/math/0607668v1

23
23

Sep 18, 2013
09/13

by
Tom Sanders

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Suppose that G is an abelian group and A is a finite subset of G containing no three-term arithmetic progressions. We show that |A+A| >> |A|(log |A|)^{1/3-\epsilon} for all \epsilon>0.

Source: http://arxiv.org/abs/math/0611304v2

7
7.0

Sep 17, 2013
09/13

by
Tom Sanders

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We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.

Source: http://arxiv.org/abs/0912.0306v2

10
10.0

Sep 20, 2013
09/13

by
Tom Sanders

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Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.

Source: http://arxiv.org/abs/1002.1552v2

7
7.0

Sep 23, 2013
09/13

by
Tom Sanders

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ECM survey article discussing the structure of subsets of Abelian groups which behave `a bit like' cosets (of subgroups).

Source: http://arxiv.org/abs/1212.0456v1

13
13

Sep 17, 2013
09/13

by
Tom Sanders

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We develop a version of Freiman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates. Our work complements three...

Source: http://arxiv.org/abs/0912.0305v2

14
14

Jul 24, 2013
07/13

by
Tom Sanders

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Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then either there is a subgroup H of G such that |A \triangle H| = o(|A|), or there is a character X on G such that -wh{1_A}(X)>> |A|^c.

Source: http://arxiv.org/abs/0807.5104v2

14
14

Sep 18, 2013
09/13

by
Tom Sanders

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We show that if A is a subset of F_2^n and |A+A| < K|A| then A is contained in a subspace of size at most 2^{O(K^{3/2}log K)}|A|. This improves on the previous best of 2^{O(K^2)}.

Source: http://arxiv.org/abs/math/0605523v2

17
17

Sep 21, 2013
09/13

by
Tom Sanders

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We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.

Source: http://arxiv.org/abs/1011.0104v3

22
22

Sep 17, 2013
09/13

by
Tom Sanders

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Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the non-abelian idempotent theorem.

Source: http://arxiv.org/abs/0912.0308v2

4
4.0

Sep 17, 2013
09/13

by
Tom Sanders

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We provide further explanation of the significance of a construction in a recent paper of Wolf [Israel J. Math. 179 (2010), 253-278] in the context of the problem of finding large subspaces in sumsets.

Source: http://arxiv.org/abs/0807.5106v2

15
15

Sep 17, 2013
09/13

by
Tom Sanders

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We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with density at least 1/2-o(1) then A+A contains a subspace of co-dimension o(n).

Source: http://arxiv.org/abs/1003.5649v2

3
3.0

Sep 18, 2013
09/13

by
Ben Green; Tom Sanders

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Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4))) indicator functions of subgroups of F_2^n.

Source: http://arxiv.org/abs/math/0605524v2

13
13

Sep 18, 2013
09/13

by
Ben Green; Tom Sanders

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Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure \mu in M(G) is said to be idempotent if \mu * \mu = \mu, or alternatively if the Fourier-Stieltjes transform \mu^ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure \mu is idempotent if and only if the set {r in G^ : \mu^(r) = 1} belongs to the coset ring of G^, that is to say we may write \mu^ as...

Source: http://arxiv.org/abs/math/0611286v2

9
9.0

Sep 23, 2013
09/13

by
Imre Z. Ruzsa; Tom Sanders

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Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.

Source: http://arxiv.org/abs/0710.0644v2